If a lease fails to specify the terms of the tenancy, then the tenancy will likely be a periodic tenancy. (CCP § 1944.) A periodic tenancy continues from one period to the next automatically unless either the landlord or tenant terminates it, at the end of the period, through proper notice. In other words, if the tenant pays rent on a monthly basis, then the tenant has a month-to-month tenancy.
Under a month-to-month tenancy, the landlord can terminate the tenancy for any reason upon thirty days written notice. (Civ. Code § 1161(1); Civ Code § 1946.) However, there may be other requirements for rent-controlled areas. Further, if the tenant has been residing in the unit for at least one year, the landlord must serve a sixty day notice.
Obviously, this is just an introduction to the unlawful detainer process and there is a great deal more involved in any unlawful detainer.
This answer does not create an attorney-client relationship. The information in this answer is merely meant for informational purposes only and does not constitute a legal advice.
Assuming the rental is located in the City of Los Angeles: If a single family home, you will have to give a 60-Day Notice to Vacate--as the tenancy will exceed one year on June 25.
But, if the rental is a multi-unit build before October 1979, then the Los Angeles Rent Stabilization Ordinance only allows evictions "for cause" and you are stuck with the tenant is he pays the rent and does not materially violate the lease terms.